Documentation

An Arc is a directed edge in a planar graph. The type s is used to tie this arc to a particular graph.

Reverse the direcion

A dart represents a bi-directed edge. I.e. a dart has a direction, however the dart of the oposite direction is always present in the planar graph as well.

Get the twin of this dart (edge)

>>> twin (dart 0 "+1")
Dart (Arc 0) -1
>>> twin (dart 0 "-1")
Dart (Arc 0) +1

test if a dart is Positive

The world in which the graph lives

A vertex in a planar graph. A vertex is tied to a particular planar graph by the phantom type s, and to a particular world w.

A *connected* Planar graph with bidirected edges. I.e. the edges (darts) are directed, however, for every directed edge, the edge in the oposite direction is also in the graph.

The types v, e, and f are the are the types of the data associated with the vertices, edges, and faces, respectively.

The orbits in the embedding are assumed to be in counterclockwise order. Therefore, every dart directly bounds the face to its right.

lens to access the Dart Data

edgeData is just an alias for dartData

Construct a planar graph, given the darts in cyclic order around each vertex.

running time: \(O(n)\).

Construct a planar graph

running time: \(O(n)\).

Construct a planar graph from a adjacency matrix. For every vertex, all vertices should be given in counter clockwise order.

pre: No self-loops, and no multi-edges

running time: \(O(n)\).

Produces the adjacencylists for all vertices in the graph. For every vertex, the adjacent vertices are given in counter clockwise order.

Note that in case a vertex u as a self loop, we have that this vertexId occurs twice in the list of neighbours, i.e.: u : [...,u,..,u,...]. Similarly, if there are multiple darts between a pair of edges they occur multiple times.

running time: \(O(n)\)

Helper function to build the graph from JSON data

running time: \(O(n)\)

Get the number of vertices

>>> numVertices myGraph
4

Get the number of Darts

>>> numDarts myGraph
12

Get the number of Edges

>>> numEdges myGraph
6

Get the number of faces

>>> numFaces myGraph
4

Enumerate all darts

Get all darts together with their data

>>> mapM_ print $ darts myGraph
(Dart (Arc 0) -1,"a-")
(Dart (Arc 2) +1,"c+")
(Dart (Arc 1) +1,"b+")
(Dart (Arc 0) +1,"a+")
(Dart (Arc 4) -1,"e-")
(Dart (Arc 1) -1,"b-")
(Dart (Arc 3) -1,"d-")
(Dart (Arc 5) +1,"g+")
(Dart (Arc 4) +1,"e+")
(Dart (Arc 3) +1,"d+")
(Dart (Arc 2) -1,"c-")
(Dart (Arc 5) -1,"g-")

Enumerate all edges. We report only the Positive darts

Enumerate all edges with their edge data. We report only the Positive darts.

>>> mapM_ print $ edges myGraph
(Dart (Arc 2) +1,"c+")
(Dart (Arc 1) +1,"b+")
(Dart (Arc 0) +1,"a+")
(Dart (Arc 5) +1,"g+")
(Dart (Arc 4) +1,"e+")
(Dart (Arc 3) +1,"d+")

Enumerate all vertices

>>> vertices' myGraph
[VertexId 0,VertexId 1,VertexId 2,VertexId 3]

Enumerate all vertices, together with their vertex data

Enumerate all faces in the planar graph

All faces with their face data.

The tail of a dart, i.e. the vertex this dart is leaving from

running time: \(O(1)\)

The vertex this dart is heading in to

running time: \(O(1)\)

endPoints d g = (tailOf d g, headOf d g)

running time: \(O(1)\)

All edges incident to vertex v, in counterclockwise order around v.

running time: \(O(k)\), where \(k\) is the output size

All incoming edges incident to vertex v, in counterclockwise order around v.

All outgoing edges incident to vertex v, in counterclockwise order around v.

Gets the neighbours of a particular vertex, in counterclockwise order around the vertex.

running time: \(O(k)\), where \(k\) is the output size

Given a dart d that points into some vertex v, report the next dart in the cyclic order around v.

running time: \(O(1)\)

Given a dart d that points into some vertex v, report the next dart in the cyclic order around v.

running time: \(O(1)\)

Data corresponding to the endpoints of the dart

Data corresponding to the endpoints of the dart

running time: \(O(1)\)

A face

The face to the left of the dart

>>> leftFace (dart 1 "+1") myGraph
FaceId 1
>>> leftFace (dart 1 "-1") myGraph
FaceId 2
>>> leftFace (dart 2 "+1") myGraph
FaceId 2
>>> leftFace (dart 0 "+1") myGraph
FaceId 0

running time: \(O(1)\).

The face to the right of the dart

>>> rightFace (dart 1 "+1") myGraph
FaceId 2
>>> rightFace (dart 1 "-1") myGraph
FaceId 1
>>> rightFace (dart 2 "+1") myGraph
FaceId 1
>>> rightFace (dart 0 "+1") myGraph
FaceId 1

running time: \(O(1)\).

The darts bounding this face, for internal faces in clockwise order, for the outer face in counter clockwise order.

running time: \(O(k)\), where \(k\) is the output size.

Generates the darts incident to a face, starting with the given dart.

\(O(k)\), where \(k\) is the number of darts reported

The vertices bounding this face, for internal faces in clockwise order, for the outer face in counter clockwise order.

running time: \(O(k)\), where \(k\) is the output size.

Get the next edge along the face

running time: \(O(1)\).

Get the previous edge along the face

running time: \(O(1)\).

Edge Oracle:

main idea: store adjacency lists in such a way that we store an edge (u,v) either in u's adjacency list or in v's. This can be done s.t. all adjacency lists have length at most 6.

note: Every edge is stored exactly once (i.e. either at u or at v, but not both)

Given a planar graph, construct an edge oracle. Given a pair of vertices this allows us to efficiently find the dart representing this edge in the graph.

pre: No self-loops and no multi-edges!!!

running time: \(O(n)\)

Builds an edge oracle that can be used to efficiently test if two vertices are connected by an edge.

running time: \(O(n)\)

Find the edge data corresponding to edge (u,v) if such an edge exists

running time: \(O(1)\)

Test if u and v are connected by an edge.

running time: \(O(1)\)

Given a pair of vertices (u,v) returns the dart, oriented from u to v, corresponding to these vertices.

running time: \(O(1)\)

Enumerates all darts such that allDarts !! i = d = i == fromEnum d